[/quote]
,
,
.
with integer coefficients such that for any positive integer 
,
is positive and 
be a triangle with incentre 
and circumcircle 
.
be the midpoints of the arcs 
of 
respectively. Let 
be the point of 
.
and the midpoint 
of 
lie on a line.
such that, for every positive integer 
the integer
is a multiple of 
(Note that 
denotes the greatest integer less than or equal to 
For example, 
and 
)
such that 
for all 
?
with complex coefficients for which ![\[P(x^2-2)=P(x)Q(x).\]](http://latex.artofproblemsolving.com/a/6/2/a62b06890b6bbcc5c9634f6de8c7ae5fe9f37828.png)
Determine all complex numbers that could be the root of 
and 
Prove that
Prove that
such that for all 
,![\[xf(x+f(y))=(y-x)f(f(x)).\]](http://latex.artofproblemsolving.com/e/4/a/e4a3bbb8b91d2aa62d699c24df342fa59be71915.png)
.
such that for all 
,![\[
f(xf(x+y)+z) = f(z) + f(x)y + f(xf(x)).
\]](http://latex.artofproblemsolving.com/f/9/3/f93b7526ff4338eea89a1c59bc009aa001ffc15d.png)
Y by
Yay, MATHCOUNTS stuff has finally been published. Here is a nice problem from this year's test(that I sillied).
Ten fair coins are flipped simultaneously. What is the probability that the
product of the number of coins landing heads up and the number landing tails up
is at least 20? Express your answer as a common fraction.
Ten fair coins are flipped simultaneously. What is the probability that the
product of the number of coins landing heads up and the number landing tails up
is at least 20? Express your answer as a common fraction.
be the number of heads and let 
be the number of tails. The problem is asking for the probability that:
.
are the only ones that work. Now, the probability that only 
heads occur and 
tails occur is just 
because it is just 
choose 
.
which should simplify to 
.
